Báo cáo hóa học: " Common fixed point and invariant approximation in hyperbolic ordered metric spaces" pptx

RESEARCH Open Access Common fixed point and invariant approximation in hyperbolic ordered metric spaces Mujahid Abbas 1 , Mohamed Amine Khamsi 2,3 and Abdul Rahim Khan 3* * Correspondence: arahim@kfupm. edu.sa 3 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Full list of author information is available at the end of the article Abstract We prove a common fixed point theorem for four mappings defined on an ordered metric space and apply it to find new common fixed point results. The existence of common fixed points is established for two or three noncommuting mappings where T is either ordered S-contraction or ordered asymptotically S-nonexpansive on a nonempty ordered starshaped subset of a hyperbolic ordered metric space. As applications, related invariant approximation results are derived. Our results unify, generalize, and complement various known comparable results from the current literature. 2010 Mathematics Subject Classification: 47H09, 47H10, 47H19, 54H25. Keywords: Hyperbolic metric space, common fixed point, Ordered uniformly C q - commuting mapping, ordered asymptotically S-nonexpansive mapping, Best approximation 1 Introduction Metric fixed point theory has primary applications in functional analysis. The interplay between geome try of Banach spaces and fi xed point theory has been very strong and fruitful. In particular, geometric conditions on underlying spaces play a crucial role for finding solution of metric fixed point problems. Altho ugh, it has purely metric flavor, it is still a major branch of nonlinear functional analysis with close ties to Banach space geometry, see for exa mple [1-4] and referenc es mentioned therein. Se veral results regarding existence and approximation of a fixed point of a mapping rely on convexity hypotheses and geometric pro perties of the Banach spaces. Recently, Khamsi and Khan [5] studied some inequali ties in hyperbolic metric spaces, which lay founda- tion for a new mathematical field: the application of geometric theory of Banach spaces to fixed point theory. Meinardus [6] was the first to employ fixed point theorem to prove th e existence of invariant approximation in Banach spaces. Subsequently, sev eral interesting and valuable results have appeared about invariant approximations [7-9]. Existence of fixed points in ordered metric spaces was first in vestigated in 2004 by Ran and Reurings [10], and then by Nieto and Lopez [11]. In 2009, Dorić [12] proved some fixed point theorems for generalized (ψ, )-weakly contractive mappings in ordered metric spaces. Recently, Radenović and Kadelburg [13] presented a result for generalized weak contractive mappings in ordered metric spaces (see also, [14,15] and references mentioned theirin). Several authors studied the Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 © 2011 Abbas et al; licensee Springer. This is an Open Access arti cle distributed under the terms of the Creative Com mons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any mediu m, provided the original work is properly cited. problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions (e.g., [16-18,13,19]). The aim of this article is to study common fixed points of (i) four mappings on an ordered metric space (ii) ordered C q -commut- ing mappings in the frame work of hyperbolic ordered metric spaces. Some results on invariant approximation for these mappings are also established which in turn extend and strengthen various known results. 2 Preliminaries Let (X , d) b e a metric space. A path joining x Î X to y Î X is a map c from a closed interval [0, l] ⊂ ℝ to X such that c(0) = x, c(l)=y,andd(c(t), c(t’)) = |t - t’ | for all t, t’ Î [0, l]. In particular, c is an isometry and d(x, y)=l. The image of c is called a metric segment joining x and y. When i t is unique the metric segment is denoted by [x, y]. We shall denote by (1 - l)x ⊕ ly the unique point z of [x, y] which satisfies d ( x, z ) = λd ( x, y ) , and d ( z, y ) = ( 1 −λ ) d ( x, y ). Such metric spaces are usually called convex metric spaces ( see T akahashi [20] and Khan at el. [21]). Moreover, if we have for all p, x, y in X d  1 2 p ⊕ 1 2 x, 1 2 p ⊕ 1 2 y  ≤ 1 2 d(x, y) , then X is called a hyperbolic metric space. It is easy to check that in this case for all x, y, z, w in X and l Î [0, 1] d (( 1 − λ ) x ⊕λy, ( 1 −λ ) z ⊕λw ) ≤ ( 1 −λ ) d ( x, z ) + λd ( y, w ). Obviously, normed linear spaces are hyperbolic spaces [5]. As nonlinear examples one can consider Hadamard manifolds [2], the Hilbert open unit ball equippe d with the hyperbolic metric [3] and CAT(0) spaces [4]. Let X be a hyperbolic ordered metric space. Throughout this article, we assume that (1 - l)x ⊕ ly ≤ (1 - l)z ⊕ lw for all x, y, z, w in X with x ≤ z and y ≤ w.AsubsetY of X is said to be ordered convex if Y includes e very metric segment joinin g a ny two of its comparable points. The set Y is said to be an orde red q-starshaped if there exists q in Y such that Y includes every metric segment joining any of its point comparable with q. Let Y be an ordered q-starshaped subset of X and f, g : Y ® Y. Put, Y f q = {y λ : y λ =(1−λ)q ⊕λfx and λ ∈ [0, 1], q ≤ x or x ≤ q} . Set, for each x in X comparable with q in Y, d(gx, Y f q )= inf λ∈ [ 0,1 ] d(gx, y λ ) . Definition 2.1.Aselfmapf on an ordered convex subset Y of a hyperbolic ordered metric space X is said to be affine if f (( 1 − λ ) x ⊕λy ) = ( 1 −λ ) fx ⊕λf y for all comparable elements x, y Î Y , and l Î [0, 1]. Let f and g be two selfmaps on X. A point x Î X is called (1) a fixed point of f if f(x) = x;(2)coincidence point of a pair (f, g)iffx = gx;(3)common fixed point of a pair (f, g)ifx = fx = gx.Ifw = fx = gx for some x in X , then w i s called a point of coincidence Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 Page 2 of 14 of f and g.Apair(f, g)issaidtobeweaklycompatibleiff and g commute at their coincidence points. We denote the set of fixed points of f by Fix(f). Definition 2.2. Let (X, ≤) be an ordered set. A pair (f, g)onX is said: (i) weakly increasing if for all x Î X, we have fx ≤ gfx and gx ≤ fgx, ([22]) (ii) partially weakly increasing if fx ≤ gfx, for all x Î X. Remark 2.3.Apair(f, g) is weakly increasing if and only if ordered pair (f, g)and (g, f) are partially weakly increasing. Example 2.4.LetX = [0, 1] be endowed with usual ordering. Let f, g : X ® X be defined by fx = x 2 and g x = √ x . Then fx = x 2 ≤ x = gfx for all x Î X. Thus (f, g ) is par- tially weakly increasing. But g x = √ x ≤ x = fgx for x Î (0, 1). So (g, f) is not partially weakly increasing. Definition 2.5.Let(X, ≤) be an ordered set. A mapping f is call ed weak annihilator of g if fgx ≤ x for all x Î X. Example 2.6.LetX = [0, 1] be endowed with usual ordering. Define f, g : X ® X by fx = x 2 and gx = x 3 . Then fgx = x 6 ≤ x for all x Î X. Thus f is a weak annihilator of g. Definition 2.7.Let(X, ≤) be an ordered set. A selfmap f on X is called dominating map if x ≤ fx for each x in X. Example 2.8.LetX = [0, 1] be endowed with usual ordering. Let f : X ® X be defined by f x = x 1 3 . Then x ≤ x 1 3 = fx for all x Î X. Thus f is a dominating map. Example 2.9. Let X = [0, ∞) be endowed with usual ordering. Define f : X ® X by fx =  n √ x for x ∈ [0, 1), x n for x ∈ [1, ∞) , n Î N. Then for all x Î X, x ≤ fx so that f is a dominating map. Definition 2.10. Let (X, ≤) be a ordered set and f and g be selfmaps on X.Thenthe pair (f, g) is said to be order limit preserving if g x 0 ≤ f x 0 , for all sequences {x n }inX with gx n ≤ fx n and x n ® x 0 . Definition 2.11.LetX be a hyperbolic ordered metric space, Y an ordered q-starshaped subset of X, f and g be selfmaps on X and q Î Fix(g). Then f is said to be: (1) ordered g-contraction if there exists k Î (0, 1) such that d ( fx, fy ) ≤ kd ( gx, gy ); for x, y Î Y with x ≤ y. (2) ordered asymptotically S-nonexpansive if there exists a sequence {k n }, k n ≥ 1, with lim n →∞ k n = 1 such that d ( f n ( x ) , f n ( y )) ≤ k n d ( gx, gy ) for each x , y in Y with x ≤ y and each n Î N .Ifk n =1,foralln Î N ,thenf is known as ordered g-nonexpansive mapping. If g = I (identity map), then f is ordered asymptotically nonexpansive mapping; Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 Page 3 of 14 (3) R-weakly commuting if there exists a real number R > 0 such that d ( fgx, gfx ) ≤ Rd ( fx, gx ); for all x in Y. (4) ordered R-subweakly commuting [23] if there exists a real number R >0such that d(fgx, gfx) ≤ Rd(gx, Y f q ) for all x Î Y. (5) o rdered uniformly R-subweakly commuting [23] if there exists a real number R > 0 such that d(f n gx, gf n x) ≤ Rd(gx, Y f n q ) for all x Î Y. (6) orde red C q -commuting [24], if gfx = fgx for all x Î C q (f, g), wh ere C q (f, g)=U {C(g, fk):0≤ k ≤ 1} and f k x =(1-k)q ⊕ kfx. (7) ordered uniformly C q -commuting,ifgf n x = f n gx for all x Î C q (g, f n ) and n Î N. (8) uniformly asymptotically regular on Y if, for ea ch h >0, there exists N(h)=N such that d(f n x, f n+1 x)<h for all h ≥ N and all x Î Y . For other related notions of noncommuting maps, we refer to [7]; in particular, here Example 2.2 and Remark 3.10(2) provide two maps which are not C q -commut- ing. Also, uniformly C q -commuting maps on X are C q -commuting and uniformly R-subweakly commuting maps are uniformly C q -commuting but the converse state- ments do not hold, in g eneral [23,25]. Fixed point theorems in a hyperconvex metric space (an example of a convex metric space) have been established by Khamsi [26] and Park [27]. Let Y be a closed subset of an ordered metric space X.Letx Î X.Defined(x, Y )= inf{d(x, y):y Î Y, y ≤ x or x ≤ y}. If there exists an element y 0 in Y comparable with x such that d(x, y 0 )=d(x, Y ), then y 0 is called an ordered best approximation to X out of Y. We denote by P Y (x), the set of all ordered best approximation to x out of Y. The reader i nterested in the interplay of fixed points and approximation theory in nor med spaces is referred to the pioneer work of Park [28] and Singh [9]. 3 Com mon fixed point in ordered metric spaces Webeginwithacommonfixedpointtheorem for two pairs of p artially weakly increasin g functions on an ordered metric space. It may regarded as the main result of this article. Theorem 3.1. Let (X, ≤, d) be an ordered metric space. Let f, g, S, and T be selfmaps on X,(T, f) an d (S, g) be partially weakly increasing with f(X) ⊆ T(X), g(X) ⊆ S(X), and dominating maps f and g be weak annihilator of T and S, respectively. Also, for every two comparable elements x, y Î X, d ( fx, gy ) ≤ hM ( x, y ), Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 Page 4 of 14 where M(x, y)=max{d(Sx, Ty), d(fx, Sx), d(gy, Ty), d(Sx, gy)+d(fx, Ty ) 2 } (3:1) for h Î [0, 1) i s satisfied. If one of f(X), g( X), S(X), or T(X) is complete subspac e of X, then {f, S} and {g, T} have unique point of coincidence in X provided that for a nonde- creasing sequence {x n } with x n ≤ y n for all n and y n ® u implies x n ≤ u. Moreover, if {f, S} and {g, T } are weakly compatible, then f, g, S, and T have a common fixed point. Proof. For any arbitrary point x 0 in X, construct sequences {x n }and{y n }inX such that y 2n−1 = f x 2n−2 = Tx 2n−1 ≤ f Tx 2n−1 ,and y 2n = g x 2n−1 = Sx 2n ≤ g Sx 2n . Since dominating maps f and g are weak annihilator of T and S , respectively so for all n ≥ 1, x 2n−2 ≤ f x 2n−2 = Tx 2n−1 ≤ f Tx 2n−1 ≤ x 2n−1 , and x 2n−1 ≤ g x 2n−1 = Sx 2n ≤ g Sx 2n ≤ x 2n . Thus, we have x n ≤ x n+1 for all n ≥ 1. Now (3.1) gives that. d ( y 2n+1 , y 2n+2 ) =d ( fx 2n , gx 2n+1 ) ≤ hM ( x 2n , x 2n+1 ) for n = 1, 2, 3, , where M(x 2n , x 2n+1 ) =max{d(Sx 2n , Tx 2n+1 ), d(fx 2n , Sx 2n ), d(gx 2n+1 , Tx 2n+1 ), d(fx 2n , Tx 2n+1 )+d(gx 2n+1 , Sx 2n ) 2  =max{d(y 2n , y 2n+1 ), d(y 2n+1 , y 2n ), d(y 2n+2 , y 2n+1 ), d(y 2n+1 , y 2n+1 )+d(y 2n+2 , y 2n ) 2 } =max{d(y 2n , y 2n+1 ), d(y 2n+1 , y 2n+2 ), d(y 2n , y 2n+1 )+d(y 2n+1 , y 2n+2 ) 2 } =max{d ( y 2n , y 2n+1 ) ,d ( y 2n+1 , y 2n+2 ) }. Now if M(x 2n , x 2n+1 )=d(y 2n , y 2n+1 ), then d(y 2n+1 , y 2n+2 ) ≤ hd(y 2n , y 2n+1 ). And if M(x 2n , x 2n+1 )=d(y 2n+1 , y 2n+2 ), then d(y 2n+1 , y 2n+2 ) ≤ hd(y 2n+1 , y 2n+2 ) which implies that d(y 2n+1 , y 2n+2 ) = 0, and y 2n+1 = y 2n+2 . Hence d ( y n , y n+1 ) ≤ hd ( y n−1 , y n ) for n =3,4, . Therefore d(y n , y n+1 ) ≤ hd(y n−1 , x n ) ≤ h 2 d ( y n−2 , y n−1 ) ≤···≤h n d ( y 0 , y 1 ) for all n Î N. Then, for m>n, d(y n , y m ) ≤ d(y n , y n+1 )+d(y n+1 , y n+2 )+···+d(y m−1 , y m ) ≤ [h n + h n+1 + ···+ h m ]d(y 0 , y 1 ) ≤ h n 1 − h d(y 0 , y 1 ), Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 Page 5 of 14 and so d(y n , y m ) ® 0asn, m ® ∞. Hence {y n } is a Cauchy sequence. Suppose that S (X) is complete. Then there exists u in S(X), such that Sx 2n = y 2n ® u as n ® ∞. Con- sequently, we can find v in X such that Sv = u. Now we claim that fv = u.Since,x 2n-2 ≤ x 2n-1 ≤ gx 2n-1 = Sx 2-n and Sx 2n ® Sv.Sothatx 2n-1 ≤ Sv and since, Sv ≤ gSv and gSv ≤ v, implies x 2n-1 ≤ v. Consider d(fv, u) ≤ d(fv, gx 2n−1 )+d(gx 2n−1 , u) ≤ hM ( v, x 2n−1 ) +d ( gx 2n−1 , u ), where M(v, x 2n−1 )=max{d(Sv, Tx 2n−1 ), d(fv, Sv), d(gx 2n−1 , Tx 2n−1 ) , d(fv, Tx 2n−1 )+d(gx 2n−1 , Sv) 2  for all n Î N. Now we have four cases: If M(v, x 2n-1 )=d(Sv, Tx 2n-1 ), then d(fv, u) ≤ hd(Sv, Tx 2n-1 )+d(gx 2n-1, u) ® 0asn ® ∞ implies that fv = u. If M(v, x 2n-1 )=d(fv, Sv), then d(fv, u) ≤ hd( fv, Sv)+d(gx 2n-1, u). Taking limit as n ® ∞ we get d(fv, u) ≤ hd(fv, u). Since h<1, so that fv = u. If M(v, x 2n-1 )=d(gx 2n-1, Tx 2n-1 ), then d(fv, u) ≤ hd(gx 2n-1, Tx 2n-1 )+d(gx 2n-1, u) ® 0 as n ® ∞ implies that fv = u. If M(v, x 2n−1 )= d( f v, Tx 2n−1 )+d(gx 2n−1 , Sv) 2 , then d(fv, u) ≤ h [d(fv, Tx 2n−1 )+d(gx 2n−1 , Sv)] 2 +d(gx 2n−1 , u) . Taking limit as n ® ∞ we get d(fv, u) ≤ h 2 d(fv, u ) .Sinceh<1, so that fv = u. Therefore, in all the cases fv = Sv = u. Since u Î f(X) ⊂ T(X), there exists w Î X such that Tw = u. Now we shall show that gw = u.As,x 2n-1 ≤ x 2n ≤ fx 2n = Tx 2n+1 and Tx 2n+1 ® Tw and so x 2n ≤ Tw.Hence,Tw ≤ fTw and fTw ≤ w, imply x 2n ≤ w. Consider d(gw, u) ≤ d(gw, fx 2n )+d(fx 2n , u) =d(fx 2n , gw)+d(fx 2n , u) ≤ hM ( x 2n , w ) +d ( fx 2n , u ), where M(x 2n , w)=max  d(Sx 2n , Tw), d(fx 2n , Sx 2n ), d(gw, Tw), d(fx 2n , Tw )+d(gw, Sx 2n ) 2  for all n Î N. Again we have four cases: If M(x 2n, w)=d(Sx 2n, Tw), then d(gw, u) ≤ h d(Sx 2n, Tw)+d(fx 2n, u) ® 0asn ® ∞. If M(x 2n ,w)=d(fx 2n, Sx 2n ), then d(gw, u) ≤ h d(fx 2n, Sx 2n )+d(fx 2n, u) ® 0asn ® ∞. If M(x 2n, w)=d(gw, Tw), then d(gw, u) ≤ hd(gw, Tw)+d(fx 2n, u)=hd(gw, u)+ d(fx 2n, u). Taking limit as n ® ∞ we get d(gw, u) ≤ hd(gw, u) which implies that gw = u.If M(x 2n , w)= d(fx 2n , Tw)+d(gw, Sx 2n ) 2 , then Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 Page 6 of 14 d(gw, u) ≤ h d( f x 2n , Tw)+d(gw, Sx 2n ) 2 +d(fx 2n , u) ≤ h 2 [d(fx 2n , u)+d(gw, Sx 2n )] + d(fx 2n , u) . Taking limit as n ® ∞ we get d(gw, u) ≤ h 2 d(gw, u ) which implies that gw = u.Fol- lowing the arguments similar to those given above, we obtain gw = Tw = u. Thus {f, S} and {g, T} have a unique point of coincidence in X.Now,if{f, S} and {g, T} are weakly compa tible, then fu = fSv = Sfv = Su = w 1 (say) and gu = gTw = Tgw = Tu = w2(say). Now d ( w 1 , w 2 ) =d ( fu, gu ) ≤ hM ( u, u ), where M(u, u)=max{d(Su, Tu), d(fu, Su), d(gu, Tu), d(fu, Tu)+d(gu, Su) 2 } = d ( w 1 , w 2 ) . Therefore d(w 1 , w 2 ) ≤ hd(w 1 , w 2 ) gives w 1 = w 2 . Hence f u = g u = Su = Tu . That is, u is a coincidence point of f, g, S,, and T. Now we shall show that u = gu. Since, v ≤ fv = u, d(u, gu)=d(fv, gu) ≤ hM ( v, u ) where M(v, u)=max  d(Sv, Tu), d(fv, Sv), d(gw, Tu), d(fv, Tu)+d(gu, Sv) 2  =d ( u, gu ) . Thus, d(u, gu) ≤ hd(u, gu) implies that gu = u. In similar way, we obtain fu = u. Hence, u is a common fixed point of f, g, S, and T. In the following result, we establish existence of a common fixed point for a pair of partially weakly increasing functions on an ordered metric space by using a control function r : R + ® R + . Theorem 3.2. Let (X, ≤, d) be an ordered metric space. Let f and g be R-weakly commuting selfmaps on X,(g, f) be partially weakly increasing with f(X) ⊆ g(X), dom- inating map f is weak annihilator of g. Suppose that for every two comparable ele- ments x, y Î X, d ( fx, fy ) ≤ r ( d ( gx, gy )), where r : R + ® R + is a continuous function such that r(t)<t for each t >0.If either f or g is continuous and one of f(X) or g(X) is complete subspace of X, then f and g have a common fixed point provided that for a nondecreasing sequence {x n } with x n ≤ y n for all n and y n ® u implies x n ≤ u. Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 Page 7 of 14 Proof. Let x 0 be an arbitrary point in X. Choose a point x 1 in X such that f x n = g x n+1 ≤ fg x n+1 . Since dominating map f is weak annihilator of g, so that for all n ≥ 1, x n ≤ f x n = g x n+1 ≤ fg x n+1 ≤ x n+1 . Thus, we have x n ≤ x n +1 for all n ≥ 1. Now d(fx n , fx n+1 ) ≤ r(d(gx n , gx n+1 ) ) = r(d(fx n−1 , fx n )) < d ( fx n−1 , fx n ) . Thus {d(fx n , fx n+1 )} is a decreasing sequence of positive real numbers and, therefore, tends to a limit L. We claim that L = 0. For if L > 0, the inequality d ( fx n , fx n+1 ) ≤ r ( d ( fx n−1 , fx n )) on taking limit as n ® ∞ and in the view of continuity of r yields L ≤ r(L)<L, a con- tradiction. Hence, L =0. For a given ε > 0, since r(ε) < ε, there is an integer k 0 such that d ( fx n , fx n+1 ) <ε− r ( ε ) ∀n ≥ k 0 . (3:2) For m, n Î N with m >n, we claim that d ( fx n , fx m ) <ε ∀n ≥ k 0 . (3:3) We prove inequality (3.3) by induction on m. Inequality (3.3) holds for m = n +1, using inequality (3.2) and the fact that ε - r (ε)<ε. Assume inequality (3.3) holds for m = k. For m = k + 1, we have d(fx n , fx m ) ≤ d(fx n , fx n+1 )+d(fx n+1 , fx m ) <ε− r(ε)+r(d(gx n+1 , gx m )) = ε − r(ε)+r(d(fx n , fx m−1 )) = ε − r(ε)+r(d(fx n , fx k )) <ε− r ( ε ) + r ( ε ) = ε. By induction on m, we conclude that inequality (3.3) holds for all m ≥ n ≥ k 0 . So {fx n } is a Cauchy sequence. Suppose that g(X) is a complete metric space. Hence {fx n } has a limit z in g(X). Also gx n ® z as n ® ∞. Let us suppose that the mapping f is continuous. Then ffx n ® fz and fg x n ® fz. Further, since f and g are R - weakly commuting, we have d ( fgx n , gf x n ) ≤ Rd ( fx n , gx n ). Taking limit as n ® ∞, the above inequality yields gffx n ® fz.Wenowassertthat z = fz. Otherwise, since x n ≤ fx n , so we have the inequality d ( fx n , ff x n ) ≤ r ( d ( gx n , gf x n )). Taking limit as n ® ∞ gives d(z, fz) ≤ r(d(z, fz)) < d(z, fz), a contradiction. Hence, z = fz.Asf(X) ⊆ g(X), there exists z 1 in X such that z = fz = gz 1 . Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 Page 8 of 14 Now, since fx n ≤ ffx n and ffxn ® fz = gz 1 and gz 1 ≤ fgz 1 ≤ z 1 imply fx n ≤ z 1 . Consider, d ( ff x n , fz 1 ) ≤ r ( d ( gf x n , gz 1 )) < d ( gf x n , gz 1 ). Taking limit as n ® ∞ implies that fz = fz 1 . This in turn implies that d ( fz, gz ) =d ( fgz 1 , gf z 1 ) ≤ Rd ( fz 1 , gz 1 ) =0 , i.e., z = fz = gz. Thus z is a common fixed point of f and g. The same conclusion is found when g is assumed to be continuous since continuity of g implies continuity of f. 4 Results in hyperbolic ordered metric spaces In this section, existence of commo n fixed points of ordered C q -commuting and ordered uniformly C q -commuting mappings is established in hyperbolic ordered metric spaces by utilizing the notions of ordered S-contractions and ordered asymptotically S- nonexpansive mappings. Theorem 4.1. Let Y be a nonempty closed ordered subset of a hyperbolic ordered metric space X. Let T and S be ordered R- subweakly commuting selfmaps on Y such that T(Y ) ⊂ S(Y ), cl(T(Y )) is compact, q Î Fix(S) and S(Y ) is complete and q-star- shaped wh ere e ach x in X is comparable with q. Let (T, S) be partially weakly increas- ing, order limit preservi ng and weakly compatible pair such that dominating map T is weak annihilator of S. If T is continuous, S-ordered nonexpansive and S is affine, then Fix(T) ∩ Fix(S) is nonempty provided that for a nondecreasing seq uence { x n } with x n ® u implies that x n ≤ u. Proof. Define T n : Y ® Y by T n ( x ) = ( 1 −λ n ) q ⊕λ n Tx , for each n ≥ 1, where l n Î (0, 1) with lim n →∞ λ n = 1 .ThenT n is a selfmap on Y for each n ≥ 1. Since S is ordered affine and T(Y ) ⊂ S(Y ), th erefor we obtain T n (Y ) ⊂ S (Y ). Note that, d(T n S x , ST n x ) = d((1 − λ n )q ⊕λ n TSx,(1− λ n )q ⊕λ n STx ) ≤ (1 −λ n )d(q, q)+λ n d(TSx, STx) = λ n d(TSx, STx) ≤ λ n Rd(Sx,(1− λ n )q ⊕λ n Tx) = λ n Rd ( Sx, T n x ) . This implies that the pair {T n , S}isorderedl n R-weakly commuting for each n.Also for any two comparable elements x and y in X, we get d(T n x, T n y) = d((1 − λ n )q ⊕λ n Tx,(1− λ n )q ⊕λ n Ty ) ≤ λ n d ( Tx, Ty ) ≤ λ n d ( Sx, Sy ) . Now following lines of the proof of Theorem 3.2, there exists x n in Y su ch that x n is a common fixed point of S and T n for each n ≥ 1. Note that d(x n , Tx n )=d(T n x n , Tx n ) = d((1 − λ n )q ⊕λ n Tx n , Tx n ) = ( 1 − λ n ) d ( q, Tx n ) . Since cl(T(Y )) is compact, there exists a positive integer M such that d ( x n , Tx n ) ≤ ( 1 −λ n ) M . Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 Page 9 of 14 The c ompactness of cl(T n (Y )) implies that there exists a subsequence {x k }of{x n } such that x k ® x 0 Î Y as k ® ∞. Now, d ( x 0 , Tx 0 ) ≤ d ( Tx 0 , Tx k ) +d ( Tx k , x k ) +d ( x k , x 0 ) and continuity of T give that x 0 Î Fix(T). Since, T is dominating map, therefore Sx k ≤ TSx k .AsT is weak annihilator of S and T is dominating, so TSx k ≤ x k ≤ Tx k . Thus Sx k ≤ Tx k and order limit preserving property of ( T, S) implies that Sx 0 ≤ Tx 0 = x 0 . Also x 0 ≤ Sx 0 . Consequently, Sx 0 = Tx 0 = x 0 . Hence the result follows. Theorem 4.2. LetYbeanonemptyclosedsubsetofacompletehyperbolicordered metric space X and let T and S be mappings on Y such that T (Y -{u}) ⊂ S(Y -{u}), where u Î Fix(S). Suppose that T is an S-contraction and continuous. Let (T, S) be par- tially weakly increasing, dominating maps T is weak annihilator of S. If T is continuous, and S and T are R-weakly commuting mappings on Y -{u}, then Fix(T)∩Fix(S) is none- mpty provided that for a nondecreasing sequence {x n } with x n ≤ y n for all n and y n ® u implies x n ≤ u. Proof. Similar to the proof of Theorem 3.2. Theorem 3.1 yields a common fixed point result for a pair of maps on an ordered startshaped subset Y of a hyperbolic ordered metric space as follows. Theorem 4.3. Let Y be a nonempty closed q- starshaped subset of a complete hyper- bolic ordered metric space X and let T and S be uniformly C q - commuting selfmapps on Y -{q} such that S(Y )=Y and T(Y -{q}) ⊂ S(Y-{q}), where q Î Fix(S). Let (T, S) be partially weakly increasing, order limit preserving and weakly compatible pair, domi- nating map T is weak annihilator of S, T is continuous and asymptotically S- nonex- pansive with sequence {k n }, as in Definition 2.11 (2), an d S is an affine mapping. For each n ≥ 1, define a mapping T n on Y by T n x =(1-a n )q ⊕ a n T n x, wh ere α n = λ n k n and { l n } is a sequence in (0, 1) with lim n →∞ λ n = 1 . Then for each n Î N, F (T n ) ∩ Fix(S) is nonempty provided that for a nondecreasing sequence {x n } with x n ≤ y n for all n and y n ® u implies x n ≤ u. Proof. For all x, y Î Y, we have d( T n ( x ) , T n ( y )) = d((1 − α n )q ⊕ α n T n x,(1− α n )q ⊕α n T n y ) ≤ α n d ( T n ( x ) , T n ( y )) ≤ λ n d ( Sx, Sy ) . Moreover, since T and S are uniformly C q -commuting and S is affine on Y with Sq = q, for each x Î C n (S, T ) ⊆ C q (S, T ), we have ST n x = S((1 −α n )q ⊕α n T n x)=(1− α n )q ⊕α n ST n x = ( 1 −α n ) q ⊕ α n T n Sx = T n Sx. Thus S and T n are weakly compatible for all n. Now, the result follows from Theo- rem 3.1. The above theorem leads to the following result. Theorem 4.4. LetYbeanonemptyclosedq-starshapedsubsetofahyperbolic ordered metric space X and let T and S be selmaps on Y such that S(Y )=Y and T(Y - {q}) ⊂ S(Y -{q}), q Î Fix(S). Let (T, S) be partially weakly increasing, order limit preser- ving, T is continuous, uniformly asymptotica lly regular, asymptotically S-nonexpansive Abbas et al. Fixed Point Theory and Applications 2011, 2011:25 http://www.fixedpointtheoryandapplications.com/content/2011/1/25 Page 10 of 14 [...]... Damjanovic, B, Djoric, D: Fixed point and common fixed point theorems on ordered cone metric spaces Appl Math Lett 23: 310–316 (2010) 23 Beg, I, Sahu, DR, Diwan, SD: Approximation of fixed points of uniformly R-subweakly commuting mappings J Math Anal Appl 324, 1105–1114 (2006) doi:10.1016/j.jmaa.2006.01.024 24 Hussain, N, Rhoades, BE: Cq-commuting maps and invariant approximations Fixed Point Theory Appl2006,... 467–472 (1999) doi:10.1016/S0362546X(98)00061-3 28 Park, S: Best approximations, inward sets and fixed points Progress in Approximation Theory pp 711–719.Acedemic Press, Inc (1991) doi:10.1186/1687-1812-2011-25 Cite this article as: Abbas et al.: Common fixed point and invariant approximation in hyperbolic ordered metric spaces Fixed Point Theory and Applications 2011 2011:25 Page 14 of 14 ... be a nonempty subset of a hyperbolic ordered metric space X, T, f, and g be selfmaps on X such that u is common fixed point of f, g, and T and T (∂M ∩ M) ⊂ M Suppose that f and g are continuous and affine on PM (u), q Î Fix(f ) ∩ Fix(g), and PM (u) is q-starshaped with f(PM (u)) = PM (u) = g(PM (u)) Let (T, f ) and (T, g) be partially weakly increasing, and dominating maps f and g be weak annihilator... contractions in partially ordered metric spaces Fixed Point Theory 11, 375–382 (2010) 15 Saadati, R, Vaezpour, SM, Vetro, P, Rhoades, BE: Fixed point theorems in generalized partially ordered G -metric spaces Math Comput Model 52, 797–801 (2010) doi:10.1016/j.mcm.2010.05.009 16 Amini-Harandi, A, Emami, H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary... exists xn in Y such that xn is a common fixed point of f, g and Tn The compactness of cl(T (Y )) implies that there exists a subsequence {Txk} of {Txn} such that Txk ® y as k ® ∞ Now, the definition of Tkxk gives that xk ® y and the result follows using continuity of T, f, and g 5 Invariant approximation In this section, we obtain results on best approximation as a fixed point of R-subweakly and uniformly... pp 485–405 Springer, Berlin (1981) 5 Khamsi, MA, Khan, AR: Inequalities in metric spaces with applications Nonlinear Analy 74, 4036–4045 (2011) doi:10.1016/j.na.2011.03.034 6 Meinardus, G: Invarianz bei linearn approximation Arch Ration Mech Anal 14, 301–303 (1963) 7 Akbar, F, Khan, AR: Common fixed point and approximation results for noncommuting maps on locally convex spaces Fixed Point Theory Appl... differential equations Nonlinear Anal 72, 2238–2242 (2010) doi:10.1016/j.na.2009.10.023 17 Beg, I, Abbas, M: Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition Fixed Point Theory Appl2006, 7 (Article ID 74503) 18 Harjani, J, Sadarangani, K: Fixed point theorems for weakly contractive mappings in partially ordered sets Nonlinear Anal 71, 3403–3410... Hussain, N, Jungck, G: Common fixed point and invariant approximation results for noncommuting generalized (f, g)nonexpansive maps J Math Anal Appl 321, 851–861 (2006) doi:10.1016/j.jmaa.2005.08.045 26 Khamsi, MA: KKM and Ky Fan theorems in hyperconvex metric spaces J Math Anal Appl 204, 298–306 (1996) doi:10.1006/jmaa.1996.0438 27 Park, S: Fixed point theorems in hyperconvex metric spaces Nonlinear... extends and improves Theorem 2.2 of Al-Thagafi [8] and Theorem 2.2(i) of Hussain and Jungck [25] in the setup of hyperbolic ordered metric spaces (b) Theorems 4.4 and 4.5 extend the results in [23] to more general classes of mappings defined on a hyperbolic ordered metric space (c) Theorems 5.1 and 5.2 set analogues of Theorems 2.11(i) and 2.12(i) in [25], respectively Acknowledgements The second and third... Theory Appl 14 (2009) (Article ID 207503) 8 Al-Thagafi, MA: Common fixed points and best approximation J Approx Theory 85, 318–320 (1996) doi:10.1006/ jath.1996.0045 9 Singh, SP: Application of a fixed point theorem to approximation theory J Approx Theory 25, 88–89 (1979) 10 Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some application to matrix equations Proc Amer Math . Abbas et al.: Common fixed point and invariant approximation in hyperbolic ordered metric spaces. Fixed Point Theory and Applications 2011 2011:25. Abbas et al. Fixed Point Theory and Applications. 54H25. Keywords: Hyperbolic metric space, common fixed point, Ordered uniformly C q - commuting mapping, ordered asymptotically S-nonexpansive mapping, Best approximation 1 Introduction Metric fixed point. appeared about invariant approximations [7-9]. Existence of fixed points in ordered metric spaces was first in vestigated in 2004 by Ran and Reurings [10], and then by Nieto and Lopez [11]. In 2009,